32 7 Vol 32 7 2011 7 Journal of Harbn Engneerng Unversty Jul 2011 do 10 3969 /j ssn 1006-7043 2011 07 018 150001 2 Yale PIE TE2 TP391 4 1006-7043 2011 07-0938-05 Kernel orthogonal and uncorrelated neghborhood preservaton dscrmnant embeddng algorthm LIU Guanqun WANG Qngjun ZHANG Rubo PAN Hawe College of Computer Scence and Technology Harbn Engneerng Unversty Harbn 150001 Chna Abstract In vew of the problems of nonlnear feature extracton n face recognton a new algorthm of orthogonal optmal dscrmnant vectors and a new algorthm of statstcally uncorrelated optmal dscrmnant vectors n a kernel space were proposed based on neghborhood preservaton embeddng NPE Frst nonlnear kernel mappng was used to map the face data nto an mplct feature space Then the algorthm maxmzed nter-class neghborhood scatter nformaton whle mnmzng ntra-class neghborhood scatter nformaton n the kernel space whch helped to mprove ts dscrmnant ablty Fnally the kernel orthogonal preservng dscrmnant cmbeddng KONPDE algorthm and the kernel uncorrelated neghborhood preservng dscrmnant embeddng KUNPDE algorthm were obtaned by constrantng the base vectors orthogonal and uncorrelated respectvely Also the general theorem for solvng the base vectors of the above two algorthms and the dervatons of the algorthms were specfcally ntroduced Experments on Yale and PIE demonstrate the effectveness of the algorthms and show that these algorthms can reduce the dmensons of the data and mprove the dscrmnant ablty Keywords manfold learnng face recognton embeddng algorthm kernel space neghborhood preservng embeddng NPE 3 locally lnear embeddng LLE 1 localty preservng projectons LPP 4 Laplacan egenmap LE 2 2010-04-01 863 2009AA04Z215 60803036 60975071 1980- E-mal luguanqun@ hrbeu edu cn 1963- E-mal zrbzrb@ hrbeu edu cn SVM KNPE kernel NPE 5 KLPP kernel LPP 6 LPP NPE LDE 7
7 939 LSDA 8 MFA 9 ANMM 10 NPDE 11 M w = I - W w T I - W w M b = I - W b T I - W b NPDE Deng Ca 12 LPP ONPE 13 OLSDA 14 OMFA 15 XM w X T V = λxm b X T V 3 2 NPDE 15 MFA NPDE 2 kernel orthogonal neghborhood preservng dscrmnant embeddng KONPDE kernel X X uncorrelated neghborhood preservng dscrmnant embeddng KUNPDE NPE V X = x 1 x 2 x n = v 1 v 2 v d Z = V T X Z = z 1 z 2 z n NP- DE Σ z - Σ W w j z j 2 j V = mn = Σ z - Σ W b jz j 1 NPE NPDE X = x 1 x 1 x n Y = y 1 y 2 y n V X Y Y = V T X NPE k V x 1 x 2 x n x α V W = Σ n α x = X α = 1 NPE mnσ y - Σ w j y j 2 = mn V T X I - W 2 = j mntr V T XMX T V 1 M = I - W T I - W d V T XX T V = I α 1 α 2 α d NPE KM w Kα = λkm b Kα 6 NPDE 3 NPDE V opt = mn Φ Vw Φ V b = mn VT X I - W w 2 V T X I - W b 2 = mn tr VT XM w X T V tr V T XM b X T V 2 mn VT X I - W w 2 V T X I - W b 2 = mn tr VT X M w X T V tr V T X M b X T V 4 K = k x x j = X T X 4 mn tr αt KM w Kα tr α T KM b Kα Φ V W Φ V b 2 W w 2 W b 1 V V T = I NPDE dst z z j = V T x - x j = 5 槡 x - x j T V V T x - x j = x - x j
940 32 2 Z 2 KM b K -1 KM w Kα k - KM b K -1 CT 槇 k-1 μ k-1 = 2λα k j E z - 13 E z z j - E z j = v T S t vj = 0 j 12 13 S t = Σ n I - CT 槇 k -1 Q k -1-1 T T x t - m x - m T k -1 C KM 槇 b K -1 KM w Kα k = = λkm b Kα k 14 = 1 X I - ee T /n X T X G X T 7 P = I - CT 槇 k - 1 Q k - 1-1 T T k - 1 C槇 KM b K - 1 14 PKM w Kα k = λkm b Kα k A = α 1 α 2 α d x z l = Σ n α T j K j v k T Cv 1 = v k φ T Cv 2 = = v k T Cv k - 1 = 0 α T k 槇 Cα 1 = α T k 槇 Cα 2 = = α T k 槇 Cα k - 1 = 0 C = I 槇 C = k C = S t C 槇 = KGK Yale PIE 1 2 1 α 1 KM w Kα = λkm b Kα k - 1 k x y = exp - x - y 2 /σ α T k = α T 槇 k Cα 2 = = α T 槇 k Cα k - 1 = 0 σ = 5 5 10 7 5 k Γ k α k 2KM w Kα k - 2λKM b Kα k - μ 1 槇 Cα 1 - μ k-1 槇 Cα k-1 = 0 9 9 α T 槇 j C KM b K -1 j =1 k -1 μ 1 α T 槇 j C KM b K -1 Cα 槇 1 + μ k-1 α T 槇 j C KM b K -1 Cα 槇 k-1 = 2α T j Q k - 1 = T T k - 1 槇 C KM b K -1 KM w Kα k 槇 C KM b K - 1 10 CT 槇 k - 1 k - 1 Q j = α T 槇 C KM b K - 1 Cα 槇 j T k - 1 = α 1 α k - 1 μ k - 1 = μ 1 μ k - 1 10 Q k-1 μ k-1 = 2T T k-1 槇 C KM b K -1 KM w Kα k 11 μ k - 1 μ k-1 = 2 Q k-1-1 T k-1 槇 C KM b K -1 KM w Kα k 9 KM b K - 1 12 l = 1 2 d z l z l 4 PCA PKM w Kα k = λkm b Kα k P = 32 32 I - CT 槇 k - 1 Q k - 1-1 T T 槇 k - 1 C KM b M - 1 T k - 1 = α 1 4 1 Yale α k - 1 Yale 15 165 α T KM b Kα = I 5 20 Γ k = α T k KM w Kα k - λ α T k KM b Kα k - 1 μ 1 α T 槇 k Cα 1 μ 2 α T 槇 k C - α 2 - Lμ k -1 α T 槇 k Cα k -1 8 20 1 Yale 2 NPE NPDE 1 LPP NPE NPDE ONPE Table 1 Fg 1 1 1 j = 1 Yale Some example mages from ORL database Yale Best recognton rate and dmenson of dfferent methods on Yale database /% LPP 76 90 23 NPE 75 91 26 ONPE 80 2 33 NPDE 81 12 20 KUNPDE 88 21 17 KONPDE 89 17 20
7 941 KONPDE KUNPDE Fg 3 3 PIE Some example mages from PIE database Fg 2 2 Yale Recognton rate vs dmenson on Yale database KUNPDE KONPDE LPP NPE ONPE NPDE 17 4 2 PIE PIE 68 41 368 13 43 4 3 PIE PIE 170 4 PIE 68 170 = 11 560 Fg 4 Recognton rate vs dmenson on PIE database 20 150 20 2 PIE Table 2 Best recognton rate and dmenson of dfferent 2 LPP NPE methods on PIE database NPDE ONPE 4 NPE NPDE /% KUNPDE LPP 79 81 147 KONPDE 65 NPE 2 ONPE 78 67 87 27 87 90 LPP NPE NPDE ONPE PIE NPDE 88 71 77 KUNPDE 94 73 70 ONPE KONPDE 95 12 67 LPP NPE LPP NPE ONPE 5 NPDE NPE NPDE LPP NPE NPE KUNPDE KONPDE KUNPDE KONPDE Yale PIE
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